Can Lorentz force be expressed as a function of z in a static electromagnetic field?

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Discussion Overview

The discussion revolves around the expression of the Lorentz force as a function of the variable z in a static electromagnetic field, where the electric field E and magnetic field B depend on z. Participants explore the implications of the particle's motion along the z-axis and the formulation of the Lorentz force in this context.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes an expression for the Lorentz force as a function of z, suggesting F(z)=q [(E *cross (1/v)) *cross (1/v) + B *cross (1/v)], based on a system of differential equations.
  • Another participant questions the validity of this expression, specifically the inclusion of the term 1/v, asking for clarification on its origin.
  • Further replies indicate confusion regarding the initial conditions of the particle's motion, emphasizing that while the initial velocity has only one component, it does not imply that the derivatives dx/dt and dy/dt are zero.
  • One participant expresses frustration with their understanding of the algebra involved in changing variables from time to z, indicating a struggle with the mathematical manipulation required.

Areas of Agreement / Disagreement

Participants do not reach consensus on the formulation of the Lorentz force as a function of z, with ongoing confusion and differing interpretations of the equations and their implications.

Contextual Notes

There are unresolved mathematical steps regarding the transformation of variables and the implications of the particle's motion in the context of the Lorentz force. The discussion reflects a mix of exploratory reasoning and technical challenges without clear resolutions.

1Keenan
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Hi,

I have a particle moving in a static electromagnetic field in which E and B have the following components:
E=(Ex, 0, 0)
B=(-Bx, 0, 0)
and both depend on z, namely Ex(z) and Bx(z).
The particle is moving along z with constant velocity v=(0, 0, vz).

If I want to express Lorentz force as a function of z, is it correct to write:

F(z)=q [(E *cross (1/v)) *cross (1/v) + B *cross (1/v)]?

I get this equation considering F=q(E+v *cross B) as a system of 6 differential equations:

dx/dt=0
dy/dt=0
dz/dt=vz
dvx/dt= q*Ex/m
dvy/dt=q*Bx*vz/m
dvz/dt=0

and expressing them as a function of z
 
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Hi 1Keenan! :smile:
1Keenan said:
dx/dt=0
dy/dt=0

No. :confused:
If I want to express Lorentz force as a function of z, is it correct to write:

F(z)=q [(E *cross (1/v)) *cross (1/v) + B *cross (1/v)]?

(write "x" not "*cross" :wink:)

Where do the 1/v come from? :confused:
 
tiny-tim said:
No. :confused:

Why?
My particle is moving along z and v=[0,0,vz]
of course it will have a displacement along x and y but the intial velocity has only one component.
Could you please explain your point?

tiny-tim said:
Where do the 1/v come from? :confused:



it comes from the differential equation:
dvx/dt=qEx/m
dvy/dt=qBxvz/m

I change the variable t in z... it is a bit of algebra I can write you everything if you want so you can double check my manipulation.
 
1Keenan said:
… the intial velocity has only one component.

but that doesn't mean that dx/dt = dy/dt = 0, not even initially :redface:
it comes from the differential equation:
dvx/dt=qEx/m
dvy/dt=qBxvz/m

I change the variable t in z... it is a bit of algebra I can write you everything if you want so you can double check my manipulation.

i still don't get it :redface:
 
tiny-tim said:
but that doesn't mean that dx/dt = dy/dt = 0, not even initially :redface:

What does it mean? :confused:

tiny-tim said:
i still don't get it :redface:

How do you write it down?
I don't understand what is tricky for you...
 
I was thinking, and actually I'm doing something stupid, but I'm really interested in expressing those differential equation as function of z and I'm lost in papers full of my wrong formulas...
at the moment I'm not able to calculate 1+1... :(
 

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